MORAINE, Ohio—Republicans’ last-gasp effort to stop Donald Trump from seizing their party’s presidential nomination kicked into high gear Friday, as his three remaining challengers avoided attacking one another, and their outside allies shifted money to states where each can take the most votes away from the front-runner.

In the starkest illustration of the strategy, Sen. Marco Rubio’s campaign urged Ohio supporters to back Gov. John Kasich, to keep the front-runner from capturing the winner-take-all state.

“There’s no question that John Kasich has the best chance to beat Donald Trump in Ohio,” said Alex Conant, Mr. Rubio’s communications director. “If you’re a Republican primary voter in Ohio, and you don’t want Donald Trump to be the nominee, John Kasich is your best bet.”

The Rubio team’s attempt to coordinate marks a recognition that Trump victories Tuesday in Ohio and Florida would give the New Yorker an unshakable grip on the GOP presidential nomination. The move comes a week after the party’s 2012 nominee, Mitt Romney, urged Republicans to vote strategically to block Mr. Trump’s path to the nomination.

Former Florida Gov. Jeb Bush, who ended his 2016 candidacy last month after months of sparring with Mr. Trump, met with each of the three non-Trump candidates this week in Miami, though he hasn’t offered direction on how to block Mr. Trump’s path.

Mr. Rubio himself allowed that Mr. Kasich is better positioned to win Ohio but stopped short of asking his backers to vote for someone else. “If a voter in Ohio thinks that voting for John Kasich is the best chance to stop Donald Trump there, I anticipate that’s what they will do,” Mr. Rubio said.

In recent days, Mr. Trump has been calling for Republicans to rally around his campaign. He has been talking up the increased turnout in GOP contests as evidence his campaign has the energy to win the general election.

Not all of Mr. Trump’s rivals were staying out of one another’s way. Texas Sen. Ted Cruz spent the morning at a megachurch in Orlando, Fla., rallying supporters in Mr. Rubio’s backyard to turn out for him in Tuesday’s big primary there. Mr. Cruz doesn’t need to win the Florida contest, but his presence in the state is a clear sign that the Texan wants Mr. Rubio to lose so he can finally have a one-on-one matchup with Mr. Trump.

“If you don’t want Donald Trump to be the nominee and lose to Hillary [Clinton], then I ask you to join our team,” Mr. Cruz said. “We welcome you. The water is warm.”

Still, Mr. Cruz is focusing his campaign efforts on two other big states voting on Tuesday, North Carolina and Missouri, where his campaign manager Jeff Roe is a veteran of hard-fought Republican politics. Mr. Cruz on Friday announced he will appear at three Missouri rallies Friday.

One of Mr. Cruz’s super PACs, Keep the Promise I, is airing a pair of TV ads in Illinois but in no other states. It is advertising on radio in Ohio, Illinois, Missouri and North Carolina, a spokeswoman said.

Ohio polls have shown a dead heat between Messrs. Kasich and Trump. A Fox News poll released Wednesday showed Mr. Kasich leading. CNN released a survey Friday showing Mr. Trump ahead, though both candidates were within the margin of error in both polls.

Mr. Kasich refused to reciprocate and rejected appeals to urge his Florida supporters to back Mr. Rubio there, where the home-state senator is polling second to Mr. Trump.

“If I’ve got supporters somewhere in the country and I’m on the ballot, I think they kinda ought to go for me,” Mr. Kasich told reporters after a rally in a glass factory here in Moraine. “What kind of a deal would it be for me if I told my people, ‘Don’t vote for me?’ ”

Kasich spokesman Chris Schrimpf said that “Sen. Rubio should immediately tell his super PAC to stop attacking the governor.”

But Mr. Rubio’s super PAC, Conservative Solutions PAC, is giving Mr. Kasich plenty of breathing room in Ohio. The PAC is airing TV advertising this weekend only in Florida, and in Illinois, where delegates are elected directly by voters.

For his part, Mr. Trump introduced his newest high-profile supporter on Friday, former candidate Ben Carson, who, like Mr. Trump, appealed to Republican voters eager for a candidate without political experience.

Groups opposed to Donald Trump are spending millions of dollars to flood the airwaves with attack ads to stop the Republican front-runner's momentum ahead of key contests in Florida and Ohio. Shelby Holliday takes a closer look at who's funding the barrage of attacks and whether the "Never Trump" message is sticking in WSJ's Campaign Q&A. Photo:

While the Rubio campaign sought to lead the charge to join forces against Mr. Trump, the front-runner launched a new offensive against Mr. Kasich. After buying more Ohio TV time Thursday, Mr. Trump’s campaign on Friday began airing an ad attacking Mr. Kasich for being “an absentee governor, spending most of his time elsewhere, especially Michigan,” a dig at the heated sports rivalry between the two states’ universities.

Mr. Trump also purchased Facebook ads this week that feature a video of his pregnant daughter, Ivanka Trump, saying, “I’m incredibly excited that in the state of Florida, you’re able to vote early.” She encouraged Floridians to cast ballots for her father by Saturday’s early-vote deadline.

Florida, Ohio, North Carolina, Illinois and Missouri all hold presidential primaries Tuesday, along with a caucus in the Northern Marianas Islands. Florida and Ohio, because they are in large states that award all of their delegates to the winner, have taken on more importance.

Mr. Kasich has said that if he loses Ohio he will end his campaign. Mr. Rubio hasn’t made the same commitment about Florida, but his campaign would be severely damaged if he doesn’t win there.

Mr. Trump heads into Tuesday’s contests with 458 delegates. Mr. Cruz is second with 359. Mr. Rubio has 151 and Mr. Kasich has 54. Ohio will award 66 delegates to its winner, and Florida is worth 99 delegates. Missouri, North Carolina and Illinois have a combined 193 delegates, though none of those states are winner-take-all, meaning the delegates likely will be split among the four candidates.

After largely avoiding direct attacks on Mr. Trump during Thursday’s Miami debate—the last opportunity before Tuesday’s contests—Messrs. Cruz and Rubio renewed their attacks on the New Yorker on Friday.

Mr. Cruz dispatched his new surrogate, former Hewlett-Packard Co. Chief Executive Carly Fiorina, to slam Mr. Trump as a coward for saying earlier in the day that he doesn’t wish to appear in any more debates, while dismissing the chances of Messrs. Kasich and Rubio to continue their campaigns after Tuesday.

At an Orlando stop Friday morning, Mr. Cruz attacked Mr. Trump as the embodiment of the deal-making Washington political culture.

“What happens, though, is that when people actually start examining Donald Trump’s record, they realize he is what they are angry about,” Mr. Cruz said before leaving Florida for an evening event in the Chicago suburbs.

Mr. Rubio, who on Friday appeared at three fundraisers for his lagging campaign, also spared Mr. Trump from attacks during Thursday’s debate. But outside a synagogue in West Palm Beach, Fla., he blasted Mr. Trump’s foreign-policy knowledge.

“We cannot afford to have a commander-in-chief and the next president of the U.S. who simply does not have the basic knowledge, not to mention the competency or the temperament, to lead this country at such a dangerous time,” Mr. Rubio said.